RSA Calculator — Key Generation, Encrypt & Decrypt

सूत्र

How this is calculated

RSA (Rivest–Shamir–Adleman, 1977) is the most widely deployed public-key cryptosystem. Security rests on the computational difficulty of factoring the product of two large primes. The key generation steps are: (1) choose two distinct primes p and q; (2) compute the modulus n = p × q; (3) compute Carmichael's totient λ(n) = lcm(p−1, q−1) — the modern standard, preferred over Euler's φ(n) = (p−1)(q−1) because it produces a smaller private exponent; (4) choose a public exponent e with 1 < e < λ(n) and gcd(e, λ(n)) = 1 — 65537 is the near-universal choice in practice because it is prime, coprime to almost all λ(n) values, and has a short binary representation that speeds modular exponentiation; (5) compute the private exponent d as the modular inverse of e modulo λ(n) via the extended Euclidean algorithm.

Encryption takes integer message m (0 ≤ m < n) and computes ciphertext c = m^e mod n using fast binary exponentiation. Decryption recovers the message as m = c^d mod n. The round-trip works because e and d are modular inverses: m^(e·d) ≡ m (mod n) by the Carmichael theorem.

This calculator uses JavaScript BigInt for exact arithmetic and is designed for educational use with small primes. Real RSA uses primes of 1024–4096 bits; this tool limits p and q to 1 000 000 to keep computation instant in the browser. A textbook RSA encrypt of a raw integer is also not semantically secure without proper padding (OAEP in production). The step-by-step panel shows every intermediate value so you can follow the algorithm by hand.